
Find integration of :-
(a)
(b)
(c)
(d)
Answer
484.5k+ views
Hint: Here, we need to find the value of the given integral. We will use integration by parts and trigonometric identities to simplify the given equation and find the integral of the given function. Integration is a method by which we can find summation of discrete data.
Formula Used: We will use the following formula:
1.Using integration by parts, the integral of the product of two differentiable functions of can be written as , where and are the differentiable functions of .
2.The tangent of an angle can be written as .
3.The secant of an angle can be written as .
Complete step-by-step answer:
We will integrate the simplified function using integration by parts.
Rewriting the equation, we get
Let be and be .
Therefore, by integrating by parts, we get
The derivative of a function of the form is .
Thus, we get
Simplifying the derivative, we get
Therefore, we get
The tangent of an angle can be written as .
The secant of an angle can be written as .
Rewriting the terms in the parentheses in terms of sine and cosine, we get
Simplifying the expression, we get
Therefore, we get
Applying the trigonometric identity , we get
Now, the integral of is .
Substituting in the equation, we get
Simplifying the expression, we get
The integral of a constant with respect to a variable is the product of the constant and the variable.
Therefore, we get
, where is the constant of integration
Therefore, we get the value of the integral as , where is the constant of integration.
Thus, the correct option is option (d).
Note: We simplified as . This is because the derivative of a function of the form is .
We simplified as . This is because the derivative of a function of the form is , and the derivative of a variable with respect to itself is 1.
We used the trigonometric identity . This is the trigonometric identity for the sine of the double of an angle.
Formula Used: We will use the following formula:
1.Using integration by parts, the integral of the product of two differentiable functions of
2.The tangent of an angle
3.The secant of an angle
Complete step-by-step answer:
We will integrate the simplified function using integration by parts.
Rewriting the equation, we get
Let
Therefore, by integrating
The derivative of a function of the form
Thus, we get
Simplifying the derivative, we get
Therefore, we get
The tangent of an angle
The secant of an angle
Rewriting the terms in the parentheses in terms of sine and cosine, we get
Simplifying the expression, we get
Therefore, we get
Applying the trigonometric identity
Now, the integral of
Substituting
Simplifying the expression, we get
The integral of a constant with respect to a variable is the product of the constant and the variable.
Therefore, we get
Therefore, we get the value of the integral
Thus, the correct option is option (d).
Note: We simplified
We simplified
We used the trigonometric identity
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